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arXiv:2008.04714

Controlled not connectivity in the Clifford group

Oscar Mario Perdomo, Reilly Ratcliffe

Published Aug 11, 2020• 0 citations
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Abstract

The Clifford group is the set of gates generated by the CZ gate, and the two local gates: the Hadamard and the Pi/2 phase shift gate. It is known that, for a two qubit system, the Clifford group C2 is a subgroup of order 92160 of the group of 4 by 4 unitary matrices. It is also known that the local Clifford gates LC2 is a subgroup of order 4608 of the group C2. In order to better understand the set C2, we make two matrices U1 and U2 in C2 equivalent if U_1=VU_2 for some V in LC2. We show that this equivalence relation splits C2 into 20 orbits, O1, ..., O20, each with 4608 elements. Moreover, for each orbit Oi, CZOi intersects 9 different orbits Oi1, ...,Oi9. Moreover, the intersection of Oij and CZOi has 512 matrices for each j=1,2, ..., ,9. The link https://www.youtube.com/watch?v=lcYtB2tnXFw&t=685s leads you to a YouTube video that explains the most important results in this paper.

DOI
10.48550/arxiv.2008.04714

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The Clifford group is the set of gates generated by the CZ gate, and the two local gates: the Hadamard and the Pi/2 phase shift gate. It is known that, for a two qubit system, the Clifford group C2 is a subgroup of order 92160 of the group of 4 by 4 unitary matrices. It is also known that the local Clifford gates LC2 is a subgroup of order 4608 of the group C2. In order to better understand the set C2, we make two matrices U1 and U2 in C2 equivalent if U_1=VU_2 for some V in LC2. We show that this equivalence relation splits C2 into 20 orbits, O1, ..., O20, each with 4608 elements. Moreover, for each orbit Oi, CZOi intersects 9 different orbits Oi1, ...,Oi9. Moreover, the intersection of Oij and CZOi has 512 matrices for each j=1,2, ..., ,9. The link https://www.youtube.com/watch?v=lcYtB2tnXFw&t=685s leads you to a YouTube video that explains the most important results in this paper.
The Clifford group is the set of gates generated by controlled not (CNOT) gates and the two local gates [Formula: see text] and [Formula: see text]. We will say that an [Formula: see text]-qubit state is a Clifford state if it can be prepared using Clifford gates, this is, [Formula: see text] is Clifford if [Formula: see text] where [Formula: see text] is a Clifford gate. In this paper, we study the set of all 4-qubit Clifford states. We prove that there are 293760 states, each of which has entanglement entropy equal to 0, [Formula: see text], 1, [Formula: see text], or [Formula: see text]. We also show that any pair of these states can be connected using local gates and at most 3 CNOT gates. We also study the Clifford states with real entries under the action of the subgroup [Formula: see text] of Clifford gates with real entries. This time we show that every pair of Clifford states with real entries can be connected with at most 5 CNOT gates and local gates in [Formula: see text]. To understand the action of the 12 different CNOT gates, we partition the Clifford states into orbits using the equivalence relation: two states are equivalent if they differ by a local Clifford gate. We label each orbit in such a way that it is easy to see the effect of the CNOT gates. Diagrams and tables explaining the action of the CNOT gates on all the orbits are presented in the paper. The link https://youtu.be/42MI6ks2_eU leads to a YouTube video that explains the most important results in this paper.
The Clifford group is the set of gates generated by the controlled not gates, the Hadamard gate and the P={{1,0},{0,i}} gate. We will say that a n-qubit state is a Clifford state if it can be prepared using Clifford gates. In this paper we study the set of all 4-qubit Clifford states. We prove that there are 293760 states and their entanglement entropy must be either 0, 2/3, 1, 4/3 and 5/3. We also show that any pair of these states can be connected using local gates and at most 3 CNOT gates. We achieve this by splitting the 293760 states into 18 groups where each pair of states in a group can be connected with a local Clifford gate. We then study how the different CNOT gates act on the 18 groups. We also study the Clifford states with real entries under the action of the subgroup C_R of Clifford gates with real entries. This time we show that every pair of Clifford states with real entries can be connected with at most 5 CNOT gates and local gates in C_R. The link https://youtu.be/42MI6ks2_eU leads to a YouTube video that explains the most important results in this paper.
The Clifford group is the set of gates generated by controlled-Z gates, the phase gate and the Hadamard gate. We will say that a n-qubit state is a Clifford state if it can be prepared using Clifford gates. These states are known as the stabilizer states and they arise in quantum error correction. In this paper we study the set of all 5-qubit Clifford states. By using an exhaustive method we start by confirming that there are 19388160 states. The main goal of the paper is to understand the action of the controlled-Z gates action on the 5-qubit states. With this goal in mind, we partition the Clifford states into orbits using the equivalence relation: two states are equivalent if they differ by a local Clifford gate. We show that there are 93 orbits, and we label each orbit in such a way that it is easy to see the effect of the controlled-Z gates. Diagrams and tables explaining the action of the CZ gates on all the orbits are presented in the paper. A similar work is done for the real Clifford 5-qubits states, this is, for states that can be prepared with Controlled-Z gates, the Z gate and the Hadamard gate.
One learned from Gottesman-Knill theorem that the Clifford model of quantum computing \cite{Clark07} may be generated from a few quantum gates, the Hadamard, Phase and Controlled-Z gates, and efficiently simulated on a classical computer. We employ the group theoretical package GAP\cite{GAP} for simulating the two qubit Clifford group C2\mathcal{C}_2C2​. We already found that the symmetric group S(6), aka the automorphism group of the generalized quadrangle W(2), controls the geometry of the two-qubit Pauli graph \cite{Pauligraphs}. Now we find that the {\it inner} group Inn(C2)=C2/Center(C2){Inn}(\mathcal{C}_2)=\mathcal{C}_2/{Center}(\mathcal{C}_2)Inn(C2​)=C2​/Center(C2​) exactly contains two normal subgroups, one isomorphic to Z2×4\mathcal{Z}_2^{\times 4}Z2×4​ (of order 16), and the second isomorphic to the parent A′(6)A'(6)A′(6) (of order 5760) of the alternating group A(6). The group A′(6)A'(6)A′(6) stabilizes an {\it hexad} in the Steiner system S(3,6,22)S(3,6,22)S(3,6,22) attached to the Mathieu group M(22). Both groups A(6) and A′(6)A'(6)A′(6) have an {\it outer} automorphism group Z2×Z2\mathcal{Z}_2\times \mathcal{Z}_2Z2​×Z2​, a feature we associate to two-qubit quantum entanglement.
We describe the structure of the nnn-qubit Clifford group Cn\mathcal{C}_nCn​ via Cayley graphs, whose vertices represent group elements and edges represent generators. In order to obtain the action of Clifford gates on a given quantum state, we introduce a quotient procedure. Quotienting the Cayley graph by the stabilizer subgroup of a state gives a reduced graph which depicts the state's Clifford orbit. Using this protocol for C2\mathcal{C}_2C2​, we reproduce and generalize the reachability graphs introduced in \cite{Keeler2022}. Since the procedure is state-independent, we extend our study to non-stabilizer states, including the W and Dicke states. Our new construction provides a more precise understanding of state evolution under Clifford circuit action.
Let us denote by [Formula: see text] the Clifford group (the circuit or operations generated by Hadamard, [Formula: see text] phase and the controlled-NOT gates) and by [Formula: see text] the set of qubit states that can be prepared by circuits from the Clifford group. In other words, a state [Formula: see text] if [Formula: see text] where [Formula: see text]. We will refer to states in [Formula: see text] as Clifford states. This paper studies the set of all three-qubit Clifford states. We prove that [Formula: see text] has 8640 states and if we define two states [Formula: see text] and [Formula: see text] in [Formula: see text] to be equivalent if [Formula: see text], with [Formula: see text] a local transformation in [Formula: see text], then the resulting quotient space has five orbits. More exactly, [Formula: see text] where the orbit [Formula: see text] is made up of states with entanglement entropy [Formula: see text]. For example, the first orbit [Formula: see text] contains the state [Formula: see text] and corresponds to the unentangled Clifford states. We say that [Formula: see text] is a real state if all its amplitudes [Formula: see text] are real numbers. We also say that an operator is real if all the entries of its matrix representation with respect to the computational basis are real numbers. In this paper, we also study the set of real Clifford 3 qubits and the way this set splits when we identify two real Clifford states [Formula: see text] and [Formula: see text] to be equivalent if [Formula: see text] where [Formula: see text] is a local real Clifford operator. An interesting aspect that follows from this study of Clifford states is the existence of two real Clifford states [Formula: see text] and [Formula: see text] that can be connected with a Clifford local transformation but they cannot be connected with a real Clifford local transformation. This is, the equation [Formula: see text] for [Formula: see text], does have a solution in the set of local transformations from [Formula: see text] but it does not have a solution among the local transformations from [Formula: see text] that are real. We go a little deeper and show that the equation [Formula: see text] does not have a solution for any local operation (not necessarily Clifford) whose entries are real numbers. Finally, we show how the CNOT gates act on the set of Clifford states and also in the set of real Clifford states.